Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays

Mediterr. J. Math. (2023) 20:280 https://doi.org/10.1007/s00009-023-02489-w 1660-5446/23/050001-17 published online August 18, 2023 c The Author(s) 2023  Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays Rasha O. A. Taie and Doaa A. M. Bakhit Abstract. In this work, we establish sufficient conditions of the uniform asymptotic stability (UAS) of solutions to second-order and third-order of Volterra integro-differential equations (VIDE) with delay. Here, we prove two new theorems on the UAS of the solutions of the considered VIDEs. Our approach is based on Lyapunov’s second method. Our results improve and form a complement to some known recent results in the literature. Two illustrative examples are considered to support the results and two graphs are drawn to illustrate the asymptotic stability of the zero solution for the considered numerical equations. The obtained results are new and original. Mathematics Subject Classification. 34K25, 45J99, 45M10. Keywords. VIDEs, UAS, delay differential equations (DDEs). 1. Introduction The integro-differential equations (IDEs), which combine differential and integral equations, have attracted more attention in recent years. Applications in mathematics, physics, biology, and engineering all heavily rely on IDEs. The equations known as the Volterra equations were studied in the early years of the 20th century by Italian mathematician Vito Volterra. In the 1930 s, Volterra showed that mathematical models for some seasonal diseases, e.g., influenza, are formulated as integral and differential equations. The use of VIDEs is widespread in the fields of biology, ecology, medicine, physics, and other sciences. To the best of our knowledge, it has been observed in a variety of physical applications, including the glass-forming process, heat transfer, the diffusion process generally, neutron diffusion, the coexistence of biological species with varying generation rates, and wind ripple in the desert. One of the most crucial methods for researching the qualitative characteristics of solutions to ordinary, functional, and IDEs is Lyapunov’s second 280 Page 2 of 17 R. O. A. Taie and D. A. M. Bakhit MJOM method because this method is widely recognized as an excellent tool in the study of differential equations. Theoretically, this method is quite significant, and it is used in many different applications, see [24]. Lyapunov’s second method is a sufficient condition to show the stability of systems, which means the system may still be stable even if one cannot find a Lyapunov-Krasovskii functional (LKF) candidate to conclude the system stability property. There are many interesting results have been obtained in the literature to study the behaviour of solutions for DDE by Lyapunov’s theory, see for example [4,10,15,16,22,25]. Besides, it is worth mentioning, that according to our observation from the literature, recently we found many exciting papers on the kind of VIDEs, for example [2,3,9–13,15–22]. In 2000, Zhang [25] investigated the uniform asymptotic stability for the linear scaler VIDE  t C(t − s)x(s)ds, ẋ(t) = Ax(t) + 0 where A ia a constant and C : R → R is a continuous function. In 2015, Tunç [14] studied the stability and the boundedness of the zero solution of the non-linear VIDE with delay of the form  t B(t, s)g(x(s))ds + p(t). ẋ(t) = − a(t)f (x(t)) + + t−τ Recently, in 2022, Appleby and Reynold [1] studied the asymptotic stability of the scalar linear VIDE  t ẋ(t) = − ax(t) + k(t − s)x(s)ds, t > 0, x(0) = x0 . 0 Our goal for this paper is to create the sufficient conditions for the UAS of second and third-order VIDEs with delay for the following equations  t ẍ + f1 (x)ẋ + h1 (t − s1 )v1 (x(s1 ))ds1 = 0, (1.1) 0 and ... x + f2 (ẋ)ẍ + αẋ +  t h2 (t − s2 )v2 (ẋ(s2 )) ds2 = 0, (1.2) 0 where h1 , h2 : [0, ∞) → (−∞, ∞) are continuous functions depend on the differences t − s1 , t − s2 , respectively, and L1 (0, ∞), L1 is the space of integrable Lebesgue functions, s1 , s2 are time delays with s1 , s2 ≤ t, also there exist two functions H1 , H2 : [0, ∞) → [0, ∞) such that Ḣ1 (t − s1 ) = d d  ∞2 (t − s2 )) = −h  ∞2 (t − s2 ) with dt∞(H1 (t − s1 )) =∞−h1 (t − s1 ), Ḣ21(t − s2 ) = dt (H |h (u)|du, |h (u)|du ∈ L [0, ∞) and |H (u)|du, |H2 (u)|du ∈ 1 2 2 0 0 t t L1 [0, ∞). The functions f1 (x), f2 (y), v1 (x) and v2 (y) are continuous scalar functions defined on R with f1 (0) = f2 (0) = v1 (0) = v2 (0) = 0. Remark 1.1. We will give the following remarks: MJOM Some New Results on the Uniform Asymptotic Page 3 of 17 280 1. Whenever, ẍ replaced by ẋ, f1 (x)ẋ replaced by Ax(t), and let v1 (x) = x(t), in the integral term then (1.1) reduces to the equation that is considered in [25]. Thus, the stability and results obtained in (1.1) include and extend the previous results. 2. In [1], If we replaced the term ẍ by ẋ, f1 (x)ẋ by ax(t), and let v1 (x) = x(t) in the integral term, then (1.1) reduces to the equation that considered in [1]. Then the stability results of this paper include and improve the stability result obtained in [1]. Then (1.1) and (1.2) generalize and improve the results obtained in [1,25]. 3. As an application in physics, many models can be modeled by IDEs. For example, first, by the Kirchhoffs second law, the net voltage drop across a closed loop equals the voltage impressed E(t). Thus, the standard closed electric RLC circuit can be governed IDE [5], second, an Abeltype Volterra integral equation describes the temperature distribution along the surface when the heat transfer to it is balanced by radiation from it. Finally, also, Abel-type Volterra integral equation determines the temperature in a semi-infinite solid, whose surface can dissipate heat by nonlinear radiation [23]. 2. Main Results Consider the general functional differential system ẋ = F (t, xt ), (2.1) where, xt represents a function from [α, t] → Rn , −∞ ≤ α ≤ t0 . For any t ≥ t0 , by (X(t), .), we shall mean the space of continuous functions φ : [α, t] → Rn , α > 0, withφ = supα≤s≤t |φ(s)|, s ∈ R and |.| is any norm on Rn . The symbol XH (t) denotes those φ ∈ X(t) with φ ≤ H for some H > 0. Here, F is a continuous function of t for t0 ≤ t ≤ ∞, whenever xt ∈ XH (t) for t0 ≤ t ≤ ∞, and takes closed bounded sets of R × X(t) into bounded sets of Rn . Theorem 2.1. [7] Let V (t, xt ) be continuous functional and locally Lipschitz for t0 ≤ t < ∞and xt ∈ XH (t). Suppose there is a continuous function Φ : [0, ∞) → [0, ∞) which is L1 [0, ∞) and satisfies   t (i) W1 (|x|) ≤ V (t, xt ) ≤ W2 (|x|) + W3 α Φ(t − s)W4 (|x(s)|)ds , where   Wi ; i = 1, 2, 3, 4 are wedges; (ii) V̇(2.1) (t, xt ) ≤ −W5 (|x|). Then, the zero solution of (2.1) is uniformly asymptotically stable (UAS). The following two theorems will be our main results for (1.1) and (1.2). 280 Page 4 of 17 R. O. A. Taie and D. A. M. Bakhit MJOM Theorem 2.2. In addition to the basic assumptions given on the functions (...truncated)